How can I make a plane's edges taper inwards?
While in Edit mode, select the plane and subdivide it, by pressing W>Subdivide. You can press F6 to adjust the number of cuts. Then, use the Proportional Editing tool, enabled by pressing O, or by going to the 3D View header, and set it's falloff to Sphere. 



Now, select the center face/vertex and scroll the mouse so the circle doesn't contain the corner vertices and scale to taste.